Recently, the author [1]} proved that \(R[G]\)  is stably finite whenever \(R\)  is left Noetherian and \(G\)  is amenable using a notion of “algebraic entropy” for modules over \(R[G]\) . Soon after, Li and Liang [2]} were able to extend this entropy theory to modules over \(R[G]\)  with \(G\)  sofic and, with this new invariant at hand, they could use similar ideas to [1]} to verify that \(R[G]\)  is stably finite whenever \(R\)  is left Noetherian and \(G\)  sofic.
In this short note, we show that the main ideas of [4]}, combined with some localization theory for Grothendieck categories, can be used to give a direct and completely self-contained proof of this last result. In fact, we obtain an even more general statement for representations of a sofic group \(G\)  on a Grothendieck category \(\mathcal {G}\)  (see Corollary REF ). The announced result then follows just specializing to the case when \(\mathcal {G}\cong R\text{-}\mathrm {Mod}\)  for a left Noetherian ring \(R\)  (see Corollary REF ).
